prod_topology — Mathlib

English

The product topology on X × Y is the infimum of the induced topologies via fst and snd.

Русский

Производная топология на X × Y равна наименьшей общности инфиксированной топологии через fst и snd.

LaTeX

$$$ (X \\boxtimes Y).str = \\operatorname{induced}(\\mathrm{Limits.prod.fst}, X.str) \\;\\inf\\; \\operatorname{induced}(\\mathrm{Limits.prod.snd}, Y.str) $$$

Lean4

theorem prod_topology {X Y : TopCat.{u}} :
    (X ⨯ Y).str = induced (Limits.prod.fst : X ⨯ Y ⟶ _) X.str ⊓ induced (Limits.prod.snd : X ⨯ Y ⟶ _) Y.str :=
  by
  let homeo := homeoOfIso (prodIsoProd X Y)
  refine homeo.isInducing.eq_induced.trans ?_
  change induced homeo (_ ⊓ _) = _
  simp [induced_compose]
  rfl

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